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Globally positive solutions of linear parabolic partial differential equations of second order with Dirichlet boundary conditions. (English) Zbl 0921.35064

The following nonautonomous parabolic partial differential equation is studied: \[ u_t= \sum^n_{i,j=1} a_{ij}(x) {\partial^2u \over\partial x_i \partial x_j}+ \sum^n_{i=1} a_i(x){\partial u\over \partial x_i}+ a_0(t,x)u, \quad t\in\mathbb{R},\;x\in\Omega, \] where \(\Omega\subset\mathbb{R}^n\) is a bounded domain and with the homogeneous boundary conditions \(u(t,x)=0\), \(t\in\mathbb{R}\), \(x\in \partial\Omega\). The main result of the paper states that the set of the solution which are defined and have constant sign for all \(t\in\mathbb{R}\) and \(x\in \Omega\), is an one-dimensional vector space. This fact is shown by applying the theory of linear skew-product semidynamical systems on Banach bundles. In the appendix generalizations to other boundary conditions are given.
Reviewer: S.Totaro (Firenze)

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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References:

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